If the equation `x^3+b x^2+c x+1=0,(bltc),` has only one real root `alpha` , then the value of `2tan^(-1)(cosec alpha)+tan^(-1)(2 sinalpha sec^2alpha)` is

If alpha in (-(pi)/(2), 0) , then find the value of tan^(-1) (cot alpha) - cot^(-1) (tan alpha)

If alpha is the only real root of the equation x^3 + bx^2 + cx + 1 = 0 (b < c) , then the value of tan^-1 alpha+ tan^-1 (alpha^-1) is equal to :

Prove that sin 4alpha = 4 tan alpha (1 - tan^(2) alpha)/((1 + tan^(2) alpha)^(2))

If the roots of the equation x^(3) -10 x + 11 = 0 are u, v, and w, then the value of 3 cosec^(2) (tan^(-1) u + tan^(-1) v + tan^(-1 w) is ____

If alpha and beta (alpha gt beta) are the roots of x^(2) + kx - 1 =0 , then find the value of tan^(-1) alpha - tan^(-1) beta

If vector vec b=(t a nalpha,-1, 2sqrt(sinalpha//2))a n d vec c=(t a nalpha, t a nalpha, 3/(sqrt(sinalpha//2))) are orthogonal and vector vec a=(1,3,sin2alpha) makes an obtuse angle with the z-axis, then the value of alpha is a alpha=tan^(-1)2 b. alpha=-tan^(-1)2 c. alpha=tan^(-1)2 d. alpha=tan^(-1)2

If alpha is a root of the equation 4x^2+2x-1=0, then prove that 4alpha^3-3alpha is the other root.

Prove that sin4 alpha=4tan alpha (1- tan^(2)alpha)/((1+tan^2 alpha)^2

If alpha,beta are roots of x^2-3x+a=0 , a in R and alpha <1< beta then find the value of a.